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We propose a nonlinear Schr\"odinger equation in a Hilbert space enlargedwith an ancilla such that the partial trace of its solution obeys to theLindblad equation of an open quantum system. The dynamics involved by thisnonlinear Schr\"odinger equation constitutes then a purification of theLindbladian dynamics. This nonlinear equation is compared with otherSchr\"odinger like equations appearing in the theory of open systems. We studythe (non adiabatic) geometric phases involved by this purification and showthat our theory unifies several definitions of geometric phases for opensystems which have been previously proposed. We study the geometry involved bythis purification and show that it is a complicated geometric structure relatedto an higher gauge theory, i.e. a categorical bibundle with a connectivestructure.

Purification of Lindblad dynamics, geometry of mixed states and geometric phases